In the paper we discuss the group theoretical algorithm
of Diffie - Hellman key exchange in the cases symmetrical
group $S_{p^n}$ and more general Cremona group of polynomial automorphisms of free module $\mathbb K^n$ over arbitrary commutative ring $\mathbb K$. We show that congagutes of affine map with nonlinear polynomial map $f$ can be element of large order and small degree. Same propertie holds
for each element of cyclic group generated by such elements. We consider some algorithms for generation of subgroups of large order and small degree of their elements.